Singular Perturbation for Beam Equation Involving Conformable Derivative: Exact and Asymptotic Analysis
Keywords:
Khalil conformable derivative, Singular perturbation, Traveling waves, Matched asymptotic expansion, Nonlinear beam equation, Boudary layer.Abstract
This work deals with an analysis of a singularly perturbed nonlinear beam equation that involves Khalil conformable derivatives. Our study is presented in two parts. We begin first by establishing the mathematical framework to derive the proposed perturbed equation and performing non-dimensionalization of the problem. For the case εα = 0, we obtain exact traveling wave solutions of the form S(ξ) = A0+A1 tanh(µξ). The second part addresses the physically relevant case εα ̸= 0 using matched asymptotic expansions. We show that the solution displays multi-scale behavior, characterized by outer regions that satisfy the reduced equation and inner boundary layers of thickness O(εα). The composite solution connects these regions and preserves the wave structure. Our analysis derives analytical expressions and establishes validity conditions, showing that the Khalil derivative introduces refined scaling.
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